NASA TM-108852: Parametric study on laminar flow for finite wings at supersonic speeds

7/23/2019 NASA TM-108852: Parametric study on laminar flow for finite wings at supersonic speeds

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NASA Technical Memorandum 108852

Parametric Study on Laminar

Flow for Finite Wings at

Supersonic Speeds

Joseph Avila Garcia, Ames Research Center, Moffett Field, California

December 1994

National Aeronautics and

Space Administration

Ames Research Center

Moffett Field, California 94035-1000


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TABLE OF CONTENTS

LIST OF FIGURES ....................................................................................................................

SUMMARY ................................................................................................................................

INTRODUCTION ......................................................................................................................

Background .....................................................................................................................

Laminar-flow control .............................................................................................

Transition ...............................................................................................................

Previous Work .................................................................................................................

Current Work ..................................................................................................................

GOVERNING EQUATIONS .....................................................................................................

Mean Flow ......................................................................................................................

Coordinate transformation .....................................................................................

Thin-layer approximation ......................................................................................

Boundary-Layer Equations .............................................................................................

Linear Stability Equations ...............................................................................................

Incompressible stability equations .........................................................................

Compressible stability equations ...........................................................................

Solution of the eigenvalue problem .......................................................................

NUMERICAL METHODS .........................................................................................................

Mean Flow ......................................................................................................................

Beam-Warming block ADI algorithm ...................................................................

Pulliam-Chaussee diagonal ADI algorithm ...........................................................

Boundary-Layer Equations .............................................................................................

Boundary-Layer Stability Equations ...............................................................................

COMPUTATIONAL GRID AND BOUNDARY CONDITIONS .............................................

Wing Grid Configurations ..............................................................................................

Wing surface generator (WSG) ................................................. . ............................

Volume grid generator ...........................................................................................

Boundary Conditions ......................................................................................................

AUTOMATED STABILITY ANALYSIS .................................................................................

RESULTS AND DISCUSSION .................................................................................................

Stability Automation Validation .....................................................................................

Reynolds Number Effects ...............................................................................................

Angle-of-Attack Effects ..................................................................................................

Page

4

4

6

9

10

11

11

13

16

17

17

18

18

18

18

19

19

19

19

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20

21

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23

iii P_liiOU_n Ni PAGE _P,,.ANIK riOT F N.ME_


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Reynolds Number Effects with Angle of Attack ..... ... .. ... ... .. ... ... ... .. ... ... ... .. ... ... ... .. ... ... ...

Sweep Effects ..................................................................................................................

CONCLUSIONS AND RECOMMENDATIONS ...... ... .. ... ... ... .. ... ... .. ... ... ... .. ... ... ... .. ... ... ... .. ... ...

Conclusions .....................................................................................................................

Recommendations ...........................................................................................................

REFERENCES ............................... .............................................................................................

APPENDIX A .............................................................................................................................

APPENDIX B .............................................................................................................................

APPENDIX C ............................................................................................................................

APPENDIX D .............................................................................................................................

24

25

26

26

27

28

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63

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LIST OF FIGURES

Figure

1.1

2.1

2.2

2.3

4.1

5.1

6.1

6.2

6.3

6.4

6.5

6.6

6.7

6.10

6.11

6.12

6.13

Page

Transition flow chart ......................................................................................................... 31

General coordinate transformation from physical to

computational space (ref. I0) ............................................................................................ 32

Boundary-layer code's coordinate system ........................................................................ 33

Disturbance wave orientation on the swept coordinate system ........................................ 34

Grid generation process .................................................................................................... 35

Automated stability analysis process ................................................................................ 36

Stability automation validation ......................................................................................... 37

Transition front result

due

to Reynolds number ............................................................... 38

Boundary-layer stability analysis region ........................................................................... 39

Chordwise pressure distribution (o_

=

0 ) . .. ... ... .. ... ... ... .. ... ... ... .. ... ... ... .. ... ... ... .. ... ... .. ... ... ... .. 40

Effect of Reynolds number on crossflow at 48% semispan .............................................. 41

Effect of Reynolds number on crossflow at 48% semispan for x/c = 10% ...................... 42

Effect of Reynolds number on shear stress in the boundary-layer at 48%

semispan for x/c

=

10% ...... ... ... .. ... ... ... .. ... ... ... .. ... ... ... .. ... ... ... .. ... ... ... .. ... ... ... .. ... ... ... .. ... ... .. .. 43

Effect of Reynolds number on transition at 48% semispan .............................................. 44

Effect

of

angle

of

attack on transition prediction at 48% semispan

for Re

=

6.34 million and 45 sweep ................................................................................ 45

Chordwise pressure distribution effects at 48% semispan due to

angle of attack at Re = 6.34 million .................................................................................. 46

Effect of angle

of

attack

on

surface flow patterns ............................................................ 47

Effect of angle of attack on leading edge flow attachment at 48% semispan ................... 48

Effect of angle of attack on crossflow profiles at 48% semispan

for Re = 6.34 million and 45 sweep ................................................................................ 49


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6.14

6.15

6.16

6.17

6.18

6.19

6.20

6.21

6.22

Maximum crossflow effect due to angle of attack at 48% semispan

for Re = 6.34 million and 45 sweep ................................................................................ 50

Effect of angle of attack on crossflow at 48% semispan

for Re = 12.68 million and 45 sweep .............................................................................. 51

Maximum crossflow effect due to angle of attack at 48% semispan

for Re = 12.68 million and 45 sweep .............................................................................. 52

Higher Reynolds number effect with angle of attack on

transition for the 45 sweep .............................................................................................. 53

Swept geometry surface grids ........................................................................................... 54

Effect of sweep on surface flow patterns at the lower

Reynolds number and 0 angle of attack case ................................................................... 55

Effect of leading edge sweep on crossflow profiles

at 48% semispan (Re = 6.34 million and t_ = 0) ............................................................. 56

Maximum crossflow effect due to leading edge sweep

at 48% semispan for Re = 12.68 million and ct = 0 ........................................................ 57

Effect of leading edge sweep on transition at 48%

semispan for Re = 12.68 million and ct = 0 .................................................................... 58

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PARAMETRIC

STUDY

ON LAMINAR FLOW FOR FINITE WINGS AT

SUPERSONIC SPEEDS

Joseph

Avila

Garcia

Ames Research Center

SUMMARY

Laminar flow

control

has been identified as a key element in

the

development of the next gen-

eration of High Speed Transports. Extending the amount of laminar flow over an aircraft will

increase range, payload, and altitude capabilities as well as lower fuel requirements, skin tempera-

ture, and therefore the overall cost. A parametric study to predict the extent of laminar flow for finite

wings at supersonic speeds was conducted using a computational fluid dynamics (CFD) code cou-

pled with a boundary layer stability code. The parameters investigated in this study were Reynolds

number, angle of attack, and sweep. The results showed that an increase in angle of attack for

specific Reynolds numbers can actually delay transition. Therefore, higher lift capability, caused by

the increased angle of attack, as well as a reduction in viscous drag, due to the delay in transition,

can be expected simultaneously. This results in larger payload and range.

INTRODUCTION

Background

Laminar flow control- Increasing the extent of laminar flow is equivalent to delaying

boundary-layer transition. This

delay

in transition

or

control

of laminar

flow is

obtained

by passive,

active, or reactive techniques (ref. 1). Passive techniques, also known as natural laminar flow (NLF)

control, are categorized as those means

of

altering the boundary-layer flow through normal aerody-

namic control parameters; for example, pressure-gradient,-wall shaping, sweep, angle of attack, and

Reynolds number.

Active techniques are categorized as those means of altering the flow through outside applied

means; for example, wall suction, heat transfer.

A third form

of

flow control is reactive flow control. Reactive flow control is the process by

which out-of-phase disturbances are artificially introduced into the boundary layer to cancel those

disturbances already present, thus stabilizing the flow and delaying transition. Some reactive con-

trols include periodic heating/cooling and wall motion. However, this method of laminar flow con-

trol is complex and, to date, is more of a theoretical method.


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The underlying principle of these techniques, as one expert puts it, is "The realization that tran-

sition is the eventual stage in a process that involves amplification of disturbances in the boundary

layer" (ref. 1).

Prediction of boundary-layer transition is an area which requires reliable methods and must be

sensitive to any control parameter that alters the mean flow. These parameters include the active,

passive, and reactive flow controls mentioned above.

Transition- The transition process is composed of several physical processes as described in

figure 1.1 (ref. 1). The transition process begins by introducing external disturbances into the bound-

ary layer through a viscous process known as receptivity (ref. 2). Some of these external distur-

bances include freestream vorticity, surface roughness, vibrations, and sound. Identifying and

defining the initialization of these external disturbances, for a given problem, is the basis for the pre-

diction of transition and creates an initial boundary-value problem. The initial disturbance is a func-

tion of the type of flow in consideration as well as its environment, and therefore is not usually

known (ref. 1).

The disturbances in the boundary layer eventually enter the critical layer and then amplify. For

low amplitude disturbances, the amplification can be modeled by linear stability theory. The normal

modes responsible for the amplification of these disturbances in the boundary-layer flow are

Tollmein-Schlichting (viscous) waves (or TS waves), Rayleigh (inflectional) waves (i.e., instabili-

ties due to crossflow or high Mach numbers), and GtJrtler vortices for curved streamlines (ref. 1).

Once the amplifications are large enough, nonlinearity sets in through secondary and tertiary

instabilities and the flow becomes transitional (ref. 1). It should be noted that the nonlinear portion

of the flow is small compared to the linear region and therefore can still often be approximated by

linear stability theory for preliminary designs.

One thing that must be avoided in all laminar flow studtes is the introduction of high levels of

initial nonlinear disturbances, which cause a bypass of the _inear disturbance regime and yield an

almost instantaneous transition. An example of such a nonlinear transition is attachment-line con-

tamination, and is commonly found in swept wings due to lhe high crossflow at the wing leading-

edge caused by turbulent flow from the fuselage.

Previous Work

Laminar flow control began in the 1930s with studies which investigated methods of natural

laminar flow (NLF) control, specifically pressure gradient tlows. This research led to the

development of the NACA 6-series airfoils in the 1940s. Natural laminar flow research was later

halted in the 1950s by the development of high speed jet ergine aircraft. These jet aircraft reached

transonic/supersonic speeds and required the wing to be swept to obtain lower local mach numbers

and maintain reasonable aircraft performance (ref. 3). The effect of sweeping the wing introduced a

three-dimensional crossflow instability that eliminated the ability to maintain laminar flow through

current existing means. The sweepback and highly favorable pressure gradient near the leading-edge

of the wing induces a boundary-layer crossflow. The sweep and adverse pressure gradient near the


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trailing-edgeikewiseinducescrossflowinstabilitieson thetrailing-edgeportionof the wing. Unlike

the more common viscous two-dimensional TS instabilities, which are damped when a favorable

pressure gradient is applied, the three-dimensional crossflow inflectional instabilities are amplified

when pressure gradients exist (ref. 4). Therefore, by reducing the presence of pressure gradient flows

over the wing, these crossflow instabilities can be reduced. One method of accomplishing this is by

using NLF airfoils which produce low pressure-gradient flows.

Natural laminar-flow control research was then replaced by attempts to actively control

boundary-layer transition, more commonly known as laminar flow control (LFC). These types of

controls are categorized as active flow control, which began with flow suction on swept wings. The

use of suction on the wing thins the boundary layer, lowering the effective Reynolds number, and

moves the crossflow boundary-layer profile closer to the high viscous wall region, damping out

crossflow instability, thus extending laminar flow (ref. 5). Work in this area peaked in the 1960s

with the flight test of the X-21A. The X-21A's work showed the basic feasibility of extending LFC

through active flow techniques at Reynolds numbers as high as 30 million (ref. 6).

Further development of the current research in LFC was delayed for a period of about ten years

due to the decreased necessity to improve aircraft fuel efficiency caused by the abundance of low

cost fuel resource and the high cost of designing such capabilities. It was not until the 1970s that

interest in LFC research was recaptured and has continued to the present day.

The need for more fuel efficient aircraft has forced aircraft designers to consider fuel efficiency a

top requirement. A major factor affecting fuel efficiency is turbulent skin friction drag. Advance-

ments in aircraft skin material manufacturing processes to include strength and smoothness, as well

as advancements in supercomputers and computing methods to analyze boundary-layer stability for

transition prediction, have made laminar-flow control a more realistic method of improving aircraft

fuel efficiency.

Turbulent skin friction drag is reduced by extending the amount of laminar flow over an aircraft.

Until recently, most studies on laminar flow have been in the subsonic flow region. Work done in

this subsonic realm has shown that turbulent skin friction drag can contribute as much as 50 percent

of the total aircraft drag (ref. 7). Studies on typical Supersonic Transports (SSTs) have shown sig-

nificant potential to increase the cruise lift-to-drag ratio by increasing the extent of laminar flow

(refs. 8 and 9). Another benefit of laminar flow at supersonic speeds includes aerodynamic heating

reduction, which allows for more skin/structure material options and, therefore, decreased aircraft

gross weight and increased range/payload capability.

Current Work

A parametric study is being conducted as an effort to numerically predict the extent of natural

laminar flow (NLF) on finite swept wings at supersonic speeds. This study is one part of the High

Speed Research Program (HSRP) underway at NASA to gain an understanding of the technical

requirements for supersonic laminar flow control (SLFC).


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As mentionedpreviously,byextendinglaminarflow overtheskin of

an aircraft, there is a signif-

icant decrease in the turbulent skin friction which, in turn, decreases the total drag force on the air-

craft's body. Furthermore, extending laminar flow at supersonic speeds will also significantly

decrease

the surface temperatures allowing for a more

optimum

selection

of

skin material.

By understanding the nature of supersonic laminar flow and the ability to control it, the follow-

ing benefits can be expected in future High Speed Research (HSR) aircraft designs: increased range,

increased payload, decreased fuel requirement, increased options for skin material, decreased initial

cost, and decreased operating cost.

The parameters that are being addressed in this study are Reynolds number, angle

of

attack, and

leading-edge wing sweep. These parameters were analyzed through the use of an advanced compu-

tational fluid dynamics (CFD) flow solver, specifically the Ames Research Center's three-

dimensional compressible Navier-Stokes (CNS) flow solver (ref. 10). From the CNS code, pressure

coefficients (Cp) are obtained for the various cases. These Cp's are then used to compute the

boundary-layer profiles through the use of the "Kaups and Cebeci" compressible boundary-layer

code (WING) (ref. 11). Finally, the boundary-layer parameters are fed into a three-dimensional

compressible boundary-layer stability code (COSAL) to predict transition (ref. 12).

The parametric study consists of a Reynolds number study, an angle-of-attack study, and a

leading-edge sweep study. The Reynolds number study addresses the Reynolds numbers of

6.34 million and 12.68 million at an angle of attack of 0 deg and leading-edge sweep of 45 deg. The

angle-of-attack study addresses the angles of attack of 0, 5, and 10 deg at the two Reynolds number

values and leading-edge sweep of 45 deg. Finally, the sweep study addresses the leading-edge

sweeps of 45 and 60 deg at the lower Reynolds number and angle of attack of 0 deg. This yields a

total of seven cases for the three studies. The above process was substantially automated through a

procedure that was developed by the work conducted under this study. This automation procedure

yields a three-dimensional graphical measure of the extent of laminar flow by predicting the transi-

tion location of laminar to turbulent flow.

GOVERNING EQUATIONS

Mean Flow

The physics of the flow in consideration can be described by the fundamental equations

governing viscous fluid flow. These fundamental equation,, are based upon the universal laws of

conservation of mass, momentum, and energy. These conservation laws are used to formulate the

time-dependent, nondimensional Navier-Stokes equations in Cartesian coordinates (X, Y, Z) as

given in the following vector form:

_gQ 3E OF 3G _Ev+3Fv4 oGv

-4-

+t _x +-_y4 +z +x "_y +z

(2.1)

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wheretheconservedquantityvector,Q, andtheEulerflux vectors,E, F, G, are:

P

pu

pv E=

pw

e

pu

2

pu +p

puv

puw

_u(e + p).

O .__

and the viscous flux vectors Ev, Fv, Gv, are:

with

[- pv

puv

= pv 2 + p

pvw

_v(e + p)

0 0

lTxx 1:xy

Ev = Re-I lyx , Fv = Re -I l:yy

l:zx

lzy

.[3x __y

G

pw

puw

pvw

pw 2 +p

w(e + p)

0

'txz

G v Re -I

=

l:y

z

'lTzz

_l]z

rxx

= A(u x + Vy + w z ) + 21.tux

ryy =

k,(u x

+

Vy

+

w z)

+ 21.try

rzz = A,(ux + Vy + w z ) + 2t.tw z

_xv = "yx = ll(Uy + Vx )

rxz = *Car= lI(u z + wx)

fyz = rZX' = I l(v: + Wy)

fix = YwPr-I 'gxeI + Urxx + vr_, + W'Cx:

fly

= y_'Pr -I

tgy,e

+

UZyx

+

VZyy

+

W_:yz

flz = )qcPr-I 3zeI + Urzx + Vrzy + wrzz

e I=e/p-0.5(u 2+v 2+w 2)

p = (y - l)[e - 0.5p(u 2 + v2 + w2)]

(2.2)

(2.3)

The variables are nondimensionalized by dividing the spatial coordinates (x, y, z) by a reference

length, L, the velocity components by the freestream speed of sound, a_, the density and viscosity

by the corresponding freestream values, and the total energy per unit volume, e, by ( pa2)_. A

5


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Newtonianfluid is assumedwith coefficientof bulkviscosity_.obtainedfromStokes'hypothesis

_,=-2/3_t.It shouldbenotedthat"y"is theratioof specificheats,"_:" is thecoefficientof thermal

conductivity,"Re" istheReynoldsnumber,and"Pr" is thePrandtlnumber.

Coordinate transformation- To solve the governing equations, it is necessary to transform

these equations into a generalized body-conforming, curvilinear coordinate system (ref. 10) as

shown in figure 2.1. This allows the development of an efficient numerical algorithm, independent

of body geometry, with a simplified application of the boundary conditions. This transformation

maps the grid points in a one-to-one correspondence with the physical points, resulting in a grid with

unit-volume cells everywhere (fig. 2.1). The general form of the transformation is expressed as:

(x,y,z) ---->(_,rl,_)

_ : _(x,y,z)

rl = rl(x,y,z) (2.4)

_ =_(x,y,z)

The chain rule of partial differentiation is applied to these transformation equations as follows:

_'--_- Xx_xx+ hx _-ff+ Zx_

_y XY_xx + hY'ffh + ZY_zz

(2.5)

az= Xz +hz +ZzTzz

where the metric terms (_x, fix, _x, %y, fly, _y, _z, nz, _z) appearing in equation 2.5 can be

determined from the following matrix differential expressions:

_q = _x fly rlz dy

_q ; ;y ;z dz

(2.6)

E x]

y = y_ yq y//_gq

(2.7)

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Therefore

{ ]

x Cy Cz x xn x

rlz = Y Yrl

x fly Y

x y z

z

zn z

yqz-yCzrl

= J]-(y_z; - y;z_)

[. YCZn- ynz

and the metric terms are represented as follows:

where J is the determinant of the Jacobian

-(XnZ -xzn)

xcz -xz

-(xczn -xnz)

Cx = J(yqz -yCzq)

Cy = J(z_x- xqz)

Cz = J(xqy-xCyq)

qx = J(zcy -yCz)

qy = J(xcz -xz)

_z = J(ycx- xCy)

Cx = J(Y@q -yqz)

Cy= J(xnz -xczn)

Cz =J(xcyq -xqy)

of the transformation

a(,n,) Cx _y _,,

- _x TIz

nx

_y _z

q

xny-xcyn ]

-(xCy-xCy)

]

Cy n -xny

(2.8)

(2.9)

(2.1 O)


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which can also be written as

where

J = l/j -1

=1/b(_'rl'_) - 1 yq Y;

_(x,y,z)

z_ z_ z;

= l/[x_(YrlZ _ - y_Zrl) - Xrl(Y_Z _ - y_z_) + x_(y_Zrl - yrlz_)]

Applying this transformation to the Navier-Stokes equations 2.1 gives

_Q _ _ _e _:_ _ _e_/

+_--_-_-_-nn-_-=_-ff_-+--_-__9_ or

(2.11)

(2.12)

P

pu

= j-I pv

9_

e

pV

puV+rlxP

pvV + fly p

pwV + rlz p

(e + p)V

9U

puU + _xP

pvU + _yp

pwU + _zP

(e + p)U

9W

puW+_xp

pvW+_yp

pwW + _zP

(e + p)W

8


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j-I

Ev=

0

_XTXX +_y_xy +_z_xz

-- _x_yx +_y_yy +_z'Cyz

_x_zx + _y'_zy +_z_zz

_xl3x +_y_y + _z_z

0

Tlxa:xx + rly'Cxy + TIz'Cxz

qx'l;yx + rlyTyy + qz_yz

rlx'Czx + rly'Czy + Tlz'CzZ

Tlx_

x +

lly_y

+ lqz_ z

0

_xa;xx +_y'_xy +_z_xz

_x_yx + _y'l:yy + _z'_yz

_x'_zx + _y_zy + _z'Czz

_x_x + _y_y + _z_z

(2.13)

where the components of the shear-stress tensor and heat-flux vector were given in equation 2.3 and

the contra-variant velocity components (U, V, W) are

U=xU+ yV+ w

V = qxU+rlyV+qzW

(2.14)

W =_xU+_yV+_zW

Thin-layer approximation- Large

amounts

of CPU time are necessary to solve the time-

dependent three-dimensional Navier-Stokes equations, particularly for flow about complex geome-

tries. To alleviate some of this large CPU requirement, a thin-layer approximation is applied to the

governing equations. This thin-layer approximation is applicable to the present study involving only

high Reynolds number flows, where the boundary layer is thin and the effects of viscosity are con-

centrated near the rigid boundaries. It should be noted that the thin-layer approximation requires that

the body surface be mapped to a coordinate surface (for the present study _ = _min) and that cluster-

ing be normal to this surface. The resulting grid has fine grid spacing in the body-normal direction

and much coarser spacing along the body. Therefore, the viscous terms in the body-normal direction

are preserved and those viscous terms in the stream and spanwise direction are neglected. This

approximation yields the following final form of the governing mean flow equations:

_)0 + c31_ _gF o3G _ 1 (OS']

03-'7 -_--_+_-t at Re I J

(2.15)

9


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where

0

+ +

_t

.(;_ +;_ +;_)v;+7(;xU;+;yV;+;zW;);y

_t

(4 2 +;_ + ;2)[0.51a(u2 +v 2 +w2); + Id (a2);]l

J Pr(7 - 1)

+_(;x u + ;yV + _zW)(;xU; +;yV; + _zW_)

and (_, i_, _" , and (_ are given by equation 2.13.

Boundary-Layer

Equations

Due to the extensive about of CPU time required to obtain an accurate boundary-layer solution,

the Navier-Stokes mean flow solution was used to provide only the pressure distribution over the

wing surface. This pressure distribution was then supplied to a boundary-layer code to provide the

boundary-layer profiles needed to predict transition. The b_mndary-layer code WING was used. This

boundary-layer code uses a conical flow approximation for the flow over a finite swept wing and

assumes a polar coordinate system as shown in figure 2.2 (ref. 11). This conical flow assumption is

valid for pressure isobars along constant percent chord lines for wings of trapezoidal planform. It

should be noted that this assumption is not valid near the tip or root of the wing due to the strong

pressure gradients created in these locations

The governing boundary-layer equations for the three-c imensional compressible laminar flow,

with the above conical flow assumption (3p/3r - 0), are given by the fundamental continuity,

momentum, and energy equations and are expressed as:

Continuity equation:

_(pru) 3 3

+ _-_ (pw) + _zz(1)rv) = 0

(2.16)

r-momentum equation:

au waU au w: 3( au'_

PU_-r+PT3--o+PV_zz-P r =_zz_ Lt-_z)

(2.17)

I0


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0-momentum equation:

3w w Ow 3w uw

OU-_--r + P r_- + PV-_z - P r

_

l _)p

3 (cOw)

r

30

+_zz[._z

)

(2.18)

Energy equation:

[.-

3H w 3H 3H 0 [ _ 3H

f'u-L-r+

07 +

Pvaz-az [Vr

0z

( l)3Iu2+w2

----I-. l- r 2

(2.19)

The following boundary conditions are then applied:

at y=O;u=O,v=Vw, W=O,(3-_-yH / = O (at the wall)

w

at y _

3;

u --) Ue, H _ H e , w _ w e (at

the

boundary-layer edge)

where y is the distance normal to the wall, and the subscript w indicates the boundary-layer quanti-

ties at the wall. The symbol 3 represents the boundary-layer thickness, and the subscript e is used to

denote boundary-layer edge quantities.

Furthermore, u is the velocity component in the radial (r) direction, w is the velocity component

normal to the radial direction, and v is the velocity component in the body-normal direction

(fig. 2.2).

Finally, it should be noted that air is treated as a perfect gas, Sutherland's law is used for [a and

the Prandtl number (Pr) is assumed constant.

Linear

Stability

Equations

The Compressible Stability AnaLysis (COSAL) code is used to analyze the stability of the three-

dimensional boundary layer (ref. 12) in order to predict transition. COSAL determines the stability

of the three-dimensional compressible Navier-Stokes equations using small-disturbance stability

theory (ref. 13). Note, that the following derivation of the linear stability equation for the compress-

ible three-dimensional flow will begin by deriving the incompressible flow (p = constant) condition

for simplicity. The derivation will be completed with the derivation of the compressible stability

equations.

Incompressible stability equations- The three-dimensional viscous incompressible flow is

expressed by the following nonlinear Navier-Stokes equations:

_U /

vV2u

--+u.

Vu=-'-Vp+

3t

p

(2.20)

11


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V.u

=0 (2.21)

The fluid motion is then decomposed into a steady flow and an instantaneous disturbance as follows:

u(x,y,x,t) = U(x,y,z) + fi(x, y, z, t) (2.22)

p(x,y,x,t) = P(x,y,z) + p(x,y,z,t) (2.23)

where, U and P are the mean flow velocities and pressures respectively in the x, y, z directions.

The x, y, z Cartesian coordinates are oriented so that x and z are the streamwise and spanwise

directions, respectively, and y is the body-normal direction. These disturbances are substituted into

equations 2.20 and 2.21. The basic terms of the original nonlinear Navier-Stokes equations are then

subtracted away, and higher powers and products of the perturbation terms, being very small, are

neglected. Finally, dynamic similitude is applied where all lengths are scaled by a reference length 1,

velocities by a reference velocity Ue, density by 9e, pressure by peu2 e, and time by 1/Ue, yielding the

following linearized disturbance equations:

-- -t- U-Vu + u. VU = -Vp+ -V2_ (2.24)

bt R

V. fi = 0 (2.25)

where R is a characteristic Reynolds number defined as:

Furthermore, a "quasiparallel" flow is assumed, which i::nplies that the mean flow is only a func-

tion of the body-normal coordinate "y" for a given point along the body. This means the velocity

only varies in the y direction and not in the x or z direction. This assumption is applicable to

boundary-layer flows since, at high Reynolds numbers, the :low gradients in the streamwise (x) or

spanwise (z) direction are much smaller than in the body-normal (y) direction. The quasiparallel

flow assumption can therefore be represented as follows:

U = U(U(y),O,W(yi)

(2.26)

where U(y) and W(y) are the velocity components in the x _nd z directions, respectively.

The linear disturbance equations are now homogeneous, separable partial differential equations

(PDEs), and the following normal mode solution applies:

12


19/108

: _exp[i(ot, x + 13z- cot)]

[*(Y)] (2.27)

(P(Y) J

where o_ and 13are the x and z components of the disturbance wave vector, k, as shown in figure 2.3

(ref. 1), and fi,O,_,_ are the complex eigenfunctions that determine the structure of the disturbance

for a given frequency (co).

Substituting equations 2.26 and 2.27 into the linearized Navier-Stokes disturbance equa-

tions 2.24 and 2.25 yields the following set of ordinary differential equations (ODEs):

]

(o_U + _W - co)fi + d___U_U= -io_ + fi(o_2 + 132) (2.28)

dy R [d2y

i(otU + [_W- w)C = -dsdP + I fd2v _(0_2 _2 ]

+ ) (2.29)

dW_,d__y_i131_ +1 [d26v - *(Or2 + _2 )] (2.30)

:

dv=0

iO_fi+ i13_v+ (2.31)

dy

Next, the following boundary conditions are applied:

at y = 0 (wall); fi(0) = 0(0) = _(0) = 0

as y --->oo (freestream); fi(y) --->0, _(y) --->0, _'(y) --->0

Note that the boundary conditions and equations 2.28 through 2.31 are homogeneous; therefore

an eigenvalue problem exists and a solution exists for only a certain combination of o_, _, and o).

This solution can be expressed by the following dispersion relation:

co = co(o_,_) (2.32)

where o_, 13, co are all complex.

Now there exists the following six arbitrary real parameters:

(Otr

,Oti ,_r ,_i

,cor,O_i )

13


20/108

which then form an eigenvalue problem.

Compressible stability equations-- The three-dimensional viscous compressible flow stability

equations are an extension of the above derived incompressible equations.

The fluid motion is decomposed into a steady flow and an instantaneous disturbance, as was

done for the incompressible nonlinear Navier-Stokes equations, as follows:

u(x,y,x,t) = U(x,y,z) + fi(x, y, z, t) (2.33)

p(x,y,x,t) = P(x,y,z) + _(x,y,z,t) (2.34)

x(x,y,x,t) = T(x,y,z) + _(x,y,z,t) (2.35)

Note that the temperature term rwas added to take into account the compressibility effects.

The Cartesian coordinate system x, y, z is used again in which the y-axis is normal to the

solid body and x, z are parallel to it. The term u, represents the x, y, z components of the instanta-

neous velocity, respectively, and p and r are the instantaneous pressure and temperature. Next,

equations 2.33 through 2.35 are substituted into the nonlinear compressible Navier-Stokes equa-

tions. The resulting equation is linearized by subtracting away the basic terms of the original non-

linear Navier-Stokes equations and neglecting higher powers and products of the perturbation terms.

Finally, assuming the basic flow is locally parallel as was done above in the quasiparallel flow

assumption of equation 2.26, the linearized compressible Navier-Stokes equations become

separable, permitting the following normal mode solution:

- ,(y)/

, _ =. _(y)texp[i(otx +

f)(Y) /

, _(Y)J

13z- cot)] (2.36)

Here, the quantities with tildas denote complex disturbance amplitudes.

Substituting equations 2.26 and 2.36 into the linearized compressible Navier-Stokes equations

yields the following system of ordinary differential equatio:ls:

(A D 2 + B D + C)_ --:0

(2.37)

14


21/108

where D represents "d/dy" and is the vector defined by

(2.38)

and A, B, C are 5 x 5 matrices given by

A_._

I 0 0 0 0

0 1 0 0 0

0 0 0 0 0

0 0 0 1 0

0 0 0 0 1

B

1 dBo

,

go dTo T6

i(K - I)(0 2 + [_2)

0

1 dg o (otU + _W6)

]-to dT o

0

i(k- 1)/_

1 dBo T'

Bo dTo

R

Bo_.

0

0 2(_- 1)M2cj(o_U_ + [3Wo)

(0_2 + [32)

1 0

0 0

2 dgo ,

Bo dTo T;

0 2(y- I)M2c(otW_ - 6U_))

(or2 + [32 )

0

0

0

1 dBo (ocW D _ [3U_))

go dTo

1 dBo

,

dToT6

15


22/108

C =

-JR (Uo+13Wo ]

_T /

+_o) xic,2+132)1J

-R , t

[--(_u o

+

f_wo)

laoTo

|+' a.o%1_2+_2)

[ _to dTo

iR/

[--{ccU o + 13Wo

_m)+(Ot2 +_2)/X ' /-'_"o f

1

.,_M2

(_U o

, f

3Wo

+co) J

-[iRoT o/Tog - 2i(y ]

(

-I)M20(aUo + 13Wg)lJ

iR / BT(7 - I)M2 1

(_u o + 13wo - co) j

1 d21ao

--[(aUo

+l_Wo).71

_o dT o I

To+I_ug+_Wo').ol|

aToJ

__L .o (_Uo+{3%)}

_-"o dTo

-l_(_U o + DW o

120 O

O 0

--m)+(R_ +ff_)--(7--

1 d_ U' 2

I)M2 --(--( o

Bo dTo

+Wo 2 ) _ d._2o (TO )2

dT o

_ dgo

T"'

dTo ol

0 0 0 0

The boundary conditions for equation 2.37 are

y = 0; 01 - (I)2= 4 = 5 = 0

y _ oo; i = 2 = 04

= 05 --")

0

l I_[ d"

(o_W_ - 13Uo) ]

Po

dTo /

+ d_ (OtWo'_[gUo)] [

dTo J

_[ iRo (ccU o +_W o]

I_T }

-o_)+(ot 2 +[32)1 J

(2.39)

(2.40)

The above boundary conditions and equations 2.37-2.39 re_resent an eigenvalue problem as was

found for the incompressible derivation represented earlier. This eigenvalue problem can also be

expressed by the dispersion relation of equation 2.23 which relates the wavenumber vector compo-

nents oc, 13with the complex frequency m. Also, note that again there exists the six arbitrary real

parameters

16


23/108

(O_ro_i,_3r,_3icor,co )

Solution of the eigenvalue problem- The eigenvalue problem can be solved by specifying

four of the six parameters mentioned above and finding the other two parameters by using

equations 2.28 2.31

for the incompressible

flow,

or

2.37 2.39

for compressible

flow.

In order to

solve the eigenvalue a temporal stability theory is used which assumes that the disturbance grows or

decays only in time

(temporally)

and not in

space

(spatially). Since o_,

[3 are

the

spatial parameters

and co is the temporal parameter (i.e., see eq. 2.36) of the disturbance, then o_, 13are assumed to be

real

and

co

complex.

Therefore, the disturbance

amplification is represented

by the

complex compo-

nent of the frequency (coi) and grows or decays as follows:

coi

> 0,

grows

coi

- ;- i >.

L.... .. .. .. .. .. .. .. .. .. .. .. . : .. .. .. .. .. .. .. : .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . _

0.002 k _ i i .. 0.002

1 F...............................................................................

L r

I

o_,,,i ......... ......

BOUNDARY LAYER PROFILE

@

X/C = 1%

0.006 ..................

o

-0.1 -0.08 -0.06 -0.04 -0.02 0 0.0:

W/U=o (Crossflow component)

BOUNDARY LAYER PROFILE @ XlC = 10%

0.006

..................

-0.1 -0.08 -0.06 -0.04 -0.02 0 O.C

W/U_

(Crossflow component)

BOUNDARY LAYER PROFILE @ X/C = 5%

0006 I .................... i,

0.004 _................................................................................ 0.004

t" i i i i 4 .-_

o.oo_....................................................... _ o.oo_

0.002 0.002

ooo,........... ...........................

0 0 '

-0.1 -0.08 -0.06 -0.04 -0.02 0 0.0 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.0;

W/Uoo (Crossflow component) W_J=o (Crossflow component)

BOUNDARY LAYER PROFILE @ X/C = 16% BOUNDARY LAYER PROFILE @ X/C = 2i%

0.006 i i i _ 0.006

...................

i

J

0.004 " i i 0.C04...........................

0.003 ; :" :: :: 4 '-_

>- _ ; ; ; ': -_ _.

0.003

0.002 .......................... i ............................................... 0.002

0.001 0.C01

0 '

0 l` ' ' E ' ' ' i ' ' ' i -' '_-_"_=_'''"

:'' ' '

-0.1 -0.08 -0.06 -0.04 -0.02 0 O.C o0.1 -0.08 -0.06 -0.04 -0.02 0 0.0_

W/Uo_ (Crossllow component) W/Uoo (Crossflow component)

Figure 6.20. Effect of leading edge sweep on crossflow profiles at 48% semispan (Re = 6.34 million and a = 0).

56


63/108

X

E

iiiiI

i

0.08 .............................................................................-........................................................

0.04

0.02

0

r-

i

i

,..-._........

.

. .... ..... .... ..... .... ..... .... ..... ..... .... ..... .... ..... .... ..... .... ..... .... ..... .... ..... .... ..... .... ..... .... ..... .... ..... .... ..... .... ... -m

Sweep

.--&.--45 deg.

_60 deg.

i

i i i i i i i i i i i i i i i i _ _ i i i

0 0.05 0.1 0.15 0.2 0.25

x/c

Figure 6.21. Maximum crossflow effect due to leading edge sweep at 48% semispan for Re

=

12.68 mil lion

and a = 0 .

5?


64/108

0.2

0.15

0.1

0.05

0

Stability Transition Curves

i I I i _ I I i i I 1 _ i

....... L____L _ ___L_J .... .............. JL__[

J

Sweep

45 deg.

60 deg.

I ..................

J

1

L__J......_L_....

L__

5000 10000 15000 20000 25000

Frequency [Hz]

Figure 6.22. Effect of leading edge sweep on transition at 48% semispan for Re = 12.68 milfion and

a --

O

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APPENDIX A

59


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67/108

WING

SURFACE GRID CREATION PROCEDURE

The following will describe the process used to generate a surface grid for any

NACA

6-

or

6a-series

airfoil.

Steps

I. Run the 6-series code "sixsefies.f" (ref. 18) with the proper input file to get an

output file called "fort. 10" containing the airfoil ordinates.

NOTE: THE FOLLOWING STEPS WILL DEPEND ON WHETHER YOU ARE

GOING TO USE VG OR S3D TO REDISTRIBUTE THE

POINTS:

II. FOR VG:

1) Use the program "airf_2dsurf.f', which will take the sixseries airfoil

ordinates output and create a file with just the upper surface ordinates of the

airfoil. The output file will be called "airf.crv".

2) Now run the code Visual Grid on the "airf.crv" file to cluster points at the

L.E. and T.E. Note, every time you redistribute the point write an output file

called " .cry" and check to see that the stretching factor is less than 1.3

[sf < 1.3]. This is done by editing the .crv file so only the newly

redistributed points are in the file and then running the program "sf.f", which

read the " .crv" file and checks each point to see if it meets the criteria of

sf < 1.3. Once the point distribution meets the criteria you now have the

output file " .crv" which is the correctly distributed upper surface airfoil

ordinates.

THINGS TO REMEMBER ABOUT REDISTRIBUTING ON VG:

3)

Specify control points at the LE and TE

Set the "SUBSET" number of points to that desired

Set the "SUBSET" point spacing to that desired

Now mna program called "conv.f" which will mirror the upper surface

ordinates from the " .crv" file as well as supply the wing surface grid

program with the needed parameters to create the surface grid. The output

file name to this program is "airfXXX.ord" (Note: XXX is the number of

points describing the airfoil)

117.FOR S3D:

1) If this is the first time redistributing the airfoil points from sixseries.f then

put the output sixseries file in the same format as the "airfXXX.ord" file

above.

2) Once this is done mn the surface grid program "WingSurf_.f which will give

a first cut to the surface grid generation. Note, use the option of MG (multi-

grid) when running the surface grid program, it will ask for this.

3) Now use the "Wingsurf_S3d.f" program which will take the upper surface of

the wing only so that it can be read into S3d.

4) Its time to use S3d to redistribute the points at LE and TE. Note the

following steps:

61

PRI__ PAC, I_I_.AN_

NOT

FILMED


68/108

Read in the file as unformatted MG Plot3d

Swap indices so you can cluster at LE & TE Which can be done by going

to[PGA] and selecting[SWAP INDICES]

To select the section to be redistribute with the mouse making sure to be

in the PICK MODE. Note, the mouse buttons give the following options:

PICK A POINT>

^PICK A LINE


69/108

APPENDIX B

63


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C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

c

C

c

c

c

C

c

c

C

c

c

c

c

C

c

c

c

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c

c

c

C

C

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c

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c

c

program WingSurf_new

Joseph A. Garcia

Date: Jan. 13, 1992

PROGRAM: This program will generate a surface grid for a

clipped delta wing with NACA 64A010 sections

using an Airfoil Potential Analytical Description

MODIFICATIONS:

MODI: To no longer use the Airfoil Description but to use as

an input from another code called sixseries.f the

normalized airfoil coordinates for a NACA 64A010, which

has been modified using Visual Grid (VG) to have

the desired chordwise point destribution. Also a span-

wise point distribution which is develop by a program

name span_dist2.f and again modified by VG to have the

desired point spacing will now be an input to this code.

MOD2: THIS IS A MOD [10/5/91] TO EXTEND THE SWEEP INTO THE

TIP SHAPE PORTION OF THE WING

MOD3: This modifies the code to allow for a taper ratio of

one with equal leading edge and trailing edge sweeps

that will now require a Aspect Ratio (AR) input.

MOD4: This mod will allow this surface grid generation code

to be able to create any sweep clipped delta wing with

out having to input a spanwise point destribution for

each 1/4 sweep and taper ratio, instead the Vinokur

stretching subroutine will be used to determine the

distribution.

MOD5: This mod is to allow the user to either input sweep

as either LE sweep or 1/4 chord sweep.

MOD6: This mod will allow this surface wing grid generation

code to be able to create any sweep wing with an

assigned aspect ratio AR which will sweep the trailing

edge of the wing as necessary.

MOD7: This mod was done to have the WingSurf_gen give the

TE_sweep for all the various wing inputs along with

the span, A_R, TR, LE_sweep, Qrt sweep as necessary.

MODS: This mod was done to sweep all of the tip zero section

of the wing with the LE_sweep.

MOD9: This mod will cluster the zero thickness trailing edge

points to match those of the swept wing.

MODI0: This mod will cluster the zero thickness Wint-Tip

section using the Vinokur streching routine and not just

mirroring the points off the wing.


72/108

c

c

c

c

c

c

c

c

c

INPUTS: Quarter-chord sweep angle

(GAMMA),

taper ratio

(lamda)

surface grid dimensions (jmax, lmax), and normalized

airfoil ordinates file named airfXXX.ord (airf127.ord

or airf200.ord).

OUTPUT: PLOT3D-format surface grid of the wing

*W*WW*W*WW****WW*WW*WW****WWWWWWWWW*WWWWW*WWWWWWWWWWWWW**WW**W**WW

parameter (jdim=500,kdim=100,1dim=10,idim=500)

dimension x(jdim, kdim, ldim),y(jdim,kdim, ldim),z(jdim, kdim, ldim),

+ x_U(idim), z_U(idim), x_L(idim), z_L(idim), yy(kdim),

+ s(150), t(100),w(50),IDM(jdim),JDM(kdim),KDM(idim)

CHARACTER*30 OUTFILE,name,INFILE

i000 FORMAT(A)

REAL GAMA, lambda, t_10, t_ll, t_12, t_13, t_14, t_15, t_21

+ , X, t_22, t_23, t_24, t_25, Chord, span, sweep, y_edg

+ , dely, delx, dely_t, delx_te, Chord_r, TE_length,Chord_t

+ , yspan, dl, d2, stotin, LE_sweep, AR, LE_length, TE_sweep

+ , Qrt_sweep, dtl, dt2, dtlt, dt2t, delwk

+ , dw0w, dwlw, dw2w, dw3w, deltp2, dely_wt, thrdspan, sf

INTEGER jmax, kmax, count, jmax_u, kmax_w, kmax_t, jmax_t

+ , counter, jmax_te, jmax_te_U, npts_U, npts_L,tr_testl

+ , imax, llmax, kk, sw_type, AR_type, tr_test2, cont_testl

+ , tmax, jj,MG, IGRID, wmax

c

c Taper ratio = lambda

c SS$$$$$$SS$$$SS$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

c SSSS$$$SSSS$$SSSSSSSSS$$$S$$$$$SSSSSS$$$SSSS$$$$$

c Qrt_sweep = 1/4 chord sweep in DEGREES

c GAMA = 1/4 chord sweep in RADS

c sweep = Leading Edge sweep in RADS

c LE_sweep = Leading Edge sweep in DEGREES

c TE_sweep = Trailing Edge sweep in DEGREES

c sweep_te = Trailing Edge sweep in RADS

c dtlt = Initial TE Wake spacing @ tip

c

dt2t = Final TE Wake spacing @ tip

c

sf = Strecthing factor (1.3)

c $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$S$$$$$$$$$$$$$$$$$

c $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

c

c

c

c

set default parameters ---

sf = 1.3

ngrid = 1

Chord_r = I. 0

dl = 0.3

d2 = 0.005

TE_length = 0.5*Chord_r

..............................................

WRITE(*, ' (a,$) ')'If you KNOW what you want your TAPER RATIO to be

+type i if NOT type 0 : '

read (*,*)tr_testl

if(tr_testl .eq. i) then

continue

else

WRITE(*,' (a,$)')'You must now specify a span since no taper was sp

+ecified (.84): '

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c

1

read (*,*)span

goto 1

endif

WRITE(*,'(a,$)')'If the taper ratio is 1 type I or 0 if not:'

read (*,*)tr test2

if(tr_test2 .eq. I) then

goto 2

else

continue

endif

WRITE(*, ' (a,$) ') ' INPUT taper rat{o: '

read (*,*) lambda

PRINT*,'If you plan to specify Aspect Ratio type 1 or 0 if not:'

read (*,*)AR_type

IF(AR_type .eq. i) THEN

WRITE(*,'(a,$)')'INPUT Aspect Ratio desired normalized by root cho

+rd: '

read (*,*)AR

if(tr_testl .ne. 1 ) then

lambda = (2*span/AR - 1.0 )

else

continue

endif

WRITE(*,'(a,$)') 'If Sweep is based on LE type I or 0 if I/4C:'

read

(*,*)sw_type

if(sw_type .eq. i) then

WRITE(*,'(a,$)') ' INPUT LE Sweep [deg]: '

read (*,*) LE_sweep

sweep= LE_sweep*(3.141592654/180)

span = AR*(l+lambda)/2

GAMA = ATAN( (span*TAN(sweep) + .25*(lambda - Chord_r))/span )

Qrt_sweep = GAMA*(180/3.141592654)

TE_sweep= ATAN( (span*TAN(sweep) - 1 + lambda)/span )

TE_sweep= TE_sweep*(180/3.141592654)

if(TE_sweep .it. 0.0 ) then

PRINT*, 'YOUR CHOICE OF INPUT YEILDS A ..... TE_SWEEP'

PRINT*,'AND THE BL CODE WING DOES NOT TAKE THIS'

PRINT*,'SO IF YOU WANT TO CONTINUE ANYWAYS TYPE 1 else 0:'

read(*,*) cont_testl

if(cont_testl .eq. 1 ) then

continue

else

PRINT*,' OK TRY AGAIN [ '

STOP

endif

else

continue

endif

else

WRITE(*, ' (a,$) ') ' INPUT 1/4 Chord Sweep [deg] : '

read (*,*) Qrt_sweep

GAMA = Qrt_sweep*(3.141592654/180)

67


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c

span = AR*(l+lambda)/2

sweep = ATAN( (span*TAN(GAMA) - .25*(lambda - Chord_r))/span )

LE_sweep= sweep*(180/3.141592654)

TE_sweep= ATAN( (span*TAN(sweep) - 1 + lambda)/span )

TE_sweep= TE_sweep*( 80/3.141592654)

endif

ELSE

WRITE(*,'(a,$)')'If Sweep is based on LE type i or 0 if i/4C:'

read (*,*)sw_type

if(sw_type .eq. i) then

WRITE(*,' (a,$)') ' INPUT LE Sweep [deg]: '

read (*,*) LE_sweep

sweep= LE_sweep*(3.141592654/180)

WRITE(*,'(a,$)')'INPUT TE_sweep if Delta wing then use 0 deg: '

read (*,*) TE_sweep

TE_sweep= TE_sweep*(3.141592654/180)


lambda = span*( TAN(TE_sweep> - TAN(sweep) ) + Chord_r

if(lambda .it. 0.0) then

PRINT*,'YOU CHOSEN TO LARGE A SPAN FOR THESE SWEEPS'

span = (0.0 - Chord_r)/(TAN(TE_sweep) -

+ TAN(sweep) )

PRINT*,'SPAN MUST BE = or > ',span

PRINT*,' TRY AGAIN '

STOP

else

continue

endif

else

continue

endif

span = (lambda -

Chord_r)/(TAN(TE

sweep) - TAN(sweep))

GAMA = ATAN( (span*TAN(sweep) + .25*(lambda - Chord_r))/span )

Qrt_sweep= GAMA*(180/3.141592654)

AR = 2*span/(l+lambda)

TE_sweep= TE_sweep*(180/3.141592654)

else

WRITE(*,'(a,$)') ' INPUT 1/4 Chord Sweep [deg]: '

read (*,*) Qrt_sweep

GAMA = Qrt_sweep*(3.141592654/180_

WRITE(*,' (a,$) ') 'INPUT TE_sweep if Delta wing then use 0 deg: '

read (*,*) TE_sweep

TE_sweep= TE_sweep*(180/3.141592654)


lambda = span*(TAN(TE_sweep) - TAN(sweep) ) + Chord_r

else

continue

endif

if( TE_sweep .eq. 0.0) then

span = (0.75 (1 - lambda))/TAN(G_A)

sweep= ATAN( (span*TAN(TE_sweep) + 1 - lambda)/span )

LE_sweep= sweep*(180/3.141592654)

TE_sweep= TE_sweep*(180/3.141592654)

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else

sweep = (.25*TAN(TE_sweep) - TAN(GAMA))/(-.75)

span= (0.25*(lambda - Chord r) )/( (TAN(GAMA) - TAN(sweep))

sweep=ATAN( (span*TAN(GAMA) - .25*(lambda - Chord_r))/span)

LE_sweep= sweep*(180/3.141592654)

TE_sweep= TE_sweep*(180/3.141592654)


endif

endif

ENDIF

PRINT *

PRINT *

PRINT *

PRINT *

PRINT *

PRINT *

'span= ' , span

'LE_sweep= ', LE_sweep

'TE_sweep= ', TE_sweep

'Qrt_sweep= ' , Qrt_sweep

AR= ,

AR

'Taper ratio= ', lambda

WRITE

read

WRITE

read

WRITE

read

*,'(a,$)')'INPUT how many point in the spanwise [25]:

*,*)kmax_w

*, '(a,$)')'INPUT initial spacing in spanwise dir. [.05]: '

* *)dl

t

*, '(a,$)')'INPUT final spacing in the spanwise dir[.005]: '

* *)d2

t

#################################################################

CALL vinokur (s, kmax_w, span, dl, d2 )

#################################################################

i=0

do 4 i=l, kmax_w

yy(i) = s(i)

k = k + 1

if(ABS(yy(i) - span) .it. 0.001) kmax_w = k

continue

#################################################################

This section will set the spanwise outer boundary

for the tip zero section.

#################################################################

MODIO

dely_wt = (yy(kmax_w) - yy(kmax_w-l) )*Chord_r

dwlw = (yy(kmax_w) - yy(kmax_w-l) )*Chord_r

Print*,' dely_wt= ',dely_wt

dw3w = 0.

wmax = 1

dw0 = dwlw

dw2w = 0.

thrdspan = 0.3*span

thrdspan = 1.0*span

do 15 jj = I,i00

deltp2 = .20*thrdspan

if(dw2w .it. deltp2) then

if(dw3w .it. thrdspan) then

dw0 = dw0*sf

wmax = wmax + 1

dw3w = dw0

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dw2w = dw3w - dw3w/sf

else

continue

endif

Continue

kmax= kmax_w + (wmax -i)

if(yy(i) .le. .5*span) then

kmax= kmax_w + (kmax_w - k)

print *, 'kmax= ' ,kmax

endif

print *, 'kmax_w= ' ,kmax_w

goto 3

#################################################################

\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

This will open the spanwise ordinate data file ceated

and then read it into an array

open (21, file= 'span2 .crv', status= 'old' ,form=' formatted' )

read (21, *)

read(21,*) kmax

k= 0

do 4 i=l,kmax

read(21,*) yy(i)

k = k + 1

if(ABS(yy(i) - span) .it. 0.001) kmax_w = k

print

*,

kmax_w= ' kmax_w

continue

goto 3

\\\\\\\\\\\\\\\\\\\\\ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

WRITE(*,'(a,$) ') ' INPUT Sweep [deg]: '

read (*,*)sweep

GAMA = sweep*(3.141592654/180)

LE_sweep = sweep

Qrt_sweep = sweep

TE_sweep = sweep

lambda = 1.0

WRITE(*, '(a,$)') ' INPUT LE or TE length [y/Cr] : '

read (*,*)LE_length

span

=

LE_Iength*COS(GAMA)

WRITE(*,'(a,$)') ' INPUT Aspect Ratio normalized by root Chord: '

read (*,*)AR

span = AR*(I + lambda)/2.0


PRINT *, 'span= ', span

PRINT *, 'LE_sweep= ', LE_sweep

PRINT *, 'Qrt_sweep= ', Qrt_swee_

PRINT *, 'TE_sweep= ', TE_sweep

PRINT *, 'AR= ', AR

WRITE(*, '(a,$)')'INPUT how many pc.int in the spanwise [kmax_w] : '

read (*,*)kmax_w

WRITE(*, ' (a,$) ') ' INPUT initial spacing in the spanwise dir. : '

read (*,*)dl

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WRITE(*,' (a, ) ) INPUT final spacing in the spanwise dir. :

read (*,*)d2

#################################################################

CALL vinokur(s,kmax_w, span,dl,d2)

#################################################################

k = 0

do 6 i=l,kmax_w

yy(i) = s(i)

k = k + 1

if(ABS(yy(i) - span) . t. 0.001) kmax_w = k

if(yy(i) .le. .5*span) then

kmax = kmax_w + (kmax_w - k)

print *, 'kmax= ' ,kmax

endif

print *,'k max_w= ' ,kmax_w

continue

#################################################################

kmax_w = 0.75*kmax

yspan = span/(kmax_w-l)

do 2 i =l,kmax

yy(i) : yspan*(i-l)

continue

\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

This will open the airfoil ordinate data file ceated

by the SIXSERIES code ref __ and then read it into an array

open(20,file='airf.ord',status='old',form='formatted')

WRITE(*,'(a,$)')' ENTER grid AIRFOIL ORDINATE INFILE NAME:

READ(*,1000)infile

open(20,file=infile,status='old',form='formatted')

read(20 i000) name

read(20

read 20

read 20

read 20

read 20

read 20

read 20

npts_U

(x_U(i),z_U(i),i=l,npts_U)

npts_L

(x_L(i),z_L(i),i=l,npts_L)

jmax te U

delx_te

TE_length

\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

jmax = npts_U + (npts_L-l)

PRINT*,'HEY jmax = ',jmax

jmax_U = npts_U

llmax = 1

dely = span/(.6*kmax-l)

kmax_w = 0.6*kmax

kmax_t = kmax - kmax_w

kmax = .2*kmax_w

MOD9a

do 50 k=l,kmax_w

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PRINT*,' k= ',k

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This will add a zero thick section behind the Wing-Trailing Edge

**** for the upper surface ****

MOD9: Starting from the Tip of the wing

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TE_sweep = ATAN( (span*TAN(sweep)-i + lambda)/span )

PRINT*,'*********** y(j,k,l)= ',y(j,k,l)

MOD9

Chord = (I + yy(k)*(TAN(TE_sweep) - TAN(sweep)) )

Chord_t = (i + yy(kmax_w)*(TAN(TE_sweep) - TAN(sweep)) )

PRINT*,' Chord= ',Chord

PRINT*,' Chord_t= ',Chord_t

delx_te = (x_U(npts_U) - x_U(npts_U-l) )*Chord

dtlt = (x_U(npts_U) - x_U(npts_U-l) )*Chord_t

PRINT*,' delx_te= ',delx_te

dtl = delx_te

dt0 = dtl

IF ( k .eq. i) THEN

PRINT*,' HI i '

tmax = 1

dt0 = dtlt

dt2t = 0.

do 7 j = i,i00

NOTE: This is sometimes change to avoid certain

conditions in Vinokur subroutine that distorts

the grid spacing.

delwk = 0.12*TE_length

delwk = 0.13*TE_length

PRINT*,' HI 2 delwk= ',delwk

if( dt2t .It. delwk) then

PRINT*,' HI 3 '

dt0 = dt0*sf

tmax

=

tmax + 1

dt3t = dt0

dt2t = dt3t - dt3t/sf

PRINT*,'#1 dtlt=',dtlt, ' dt2t=',dt2t

else

continue

endif

continue


PRINT*, '#I dtlt=',dtlt, ' dt2t=',dt2t

ELSE

CONTINUE

ENDIF


dt2 = dt2t

PRINT*,'#2 dtl=',dtl,' dt2=',dt_

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jmax te U = tmax - 1

jmax_te

=

2*jmax te U

PRINT*,'HEY 1 jmax_te= ',jmax_te

PRINT*,'dt3t= ',dt3t

PRINT*, 'tmax= ',tmax

#################################

CALL vinokur(t,tmax,TE_length,dtl,dt2)

#################################

jj = tmax + 1

if(tr_test2 .eq. i) then

TE_sweep = GAMA

else

continue

endif

do i0 j= l,jmax te U

jj = jj - 1

PRINT*, 't(jj)= ',t(jj), ' jj= ',jj

y(j,k,l) = yy(k)

x(j,k,l)

=

Chord_r + y(j,k,l)*(TAN(TE_sweep)) + t(jj)

PRINT*,' x(j,k,l)= ',x(j,k,l),' j= ',j

z(j,k,l) = 0.0

Continue

This will

compute

the upper surface of the wing

starting from the root trailing edge.

**** for the upper surface

****

MOD9

i = npts_U - jmax te U + 1

i = npts_U + 1

do 20 j=jmax te

U

+ l,jmax_U + jmax te

U

i =i - 1

y(j,k,l) = yy(k)


Chord = 1.0

x(j k,l)

=

Chord * x_U(i) + (y(j,k,I)*TAN(GAMA))

else

TE_sweep = ATAN( (span*TAN(sweep)-I + lambda)/span )

Chord = (I + y(j,k,l)*(TAN(TE_sweep) - TAN(sweep)) )

x(j,k,l)

=

Chord

*

x_U(i + y(j,k,l

*TAN(sweep)

endif

\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

z(j,k,l) = Chord

*

z_U(i)

\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

continue

\\\\\\\\\\

\\\\\\\\\\

\\\\\\\\\\\\\\\\\\\\\\\\

\\\\\\\\\\\\\\\\\\\\\\\\

This will compute the lower surface of the wing

starting from the root leading edge

count= 1

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MOD9

do 30 j=jmax_U+l,jmax - jmax te U

do 30 j=jmax_U + jmax te U + l,jmax + jmax te U

count = count + 1

y(j,k,l) = yy(k)

\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

if(tr_test2 .eq. i) then

x(j,k,l)

=

Chord * x_L(count) + (y(j,k,I)*TAN(GAMA))

else


Chord

=

(I + y(j,k,l)*(TAN(TE_sweep) - TAN(sweep)) )

x(j,k,l) = Chord * x_L(count) + y(j,k,l)*TAN(sweep)

endif

\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

This section will read in the ordinate of the airfiol and

convert it to the proper values to define the wing

\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

z(j,k,l) = Chord

*

z_L(count)

\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

continue

This will add a zero thick section behind the Wing - Trailing Edge

**** for the lower surface ****

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MOD9

jj = 1

do 40 j= jmax - jmax te U + l,jmax

do 40 j= jmax + jmax_te_U + i,jmax + 2*jmax te U

jj = jj + 1

y(j,k,l) = yy(k)


x(j,k,l) = t(jj) + y(j,k,I)*TAN(GAMA)

else

TE_sweep = ATAN( (span*TAN(sweep)-I + lambda)/span )

x(j,k,l) = Chord_r + y(j,k,l)*TAN(TE_sweep) + t(jj)

endif

z(j,k,l = 0.0

Continue

contlnue

kk = kmax_w *** MODI0

***

kk = 1

do i00 k= kmax_w+l, kmax

This will add a zero thick section cff the Wing Tip-Trailing Edge

**** for the upper surface ****

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MODI0

kk = kk -i

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dely_wt = (y(l,kmax_w,l) - y(l,kmax_w-l,l) )*Chord_r

dwlw = (y(l,kmax_w,l) - y(l,kmax_w-l,l) )*Chord_r

Print*,' dely_wt= ',dely_wt

wmax = 1

dw0 = dwlw

dw3w = 0.

dw2w = 0.

thrdspan = 0.3*span

thrdspan = 1.0*span

do 55 jj = i,i00

deltp2 = .2*thrdspan

if(dw2w .it. deltp2) then

if(dw3w .it. thrdspan) then

dw0 = dw0*sf

wmax = wmax + 1

dw3w = dw0


else

continue

endif

Continue


dtl = dely_wt

dt2 = deltp2

dt2 = dw2w

#################################

CALL vinokur(w,wmax,thrdspan,dtl,dt2)

#################################

kk

=

kk + 1

MOD9

i = npts_U + 1

jj = tmax + 1

do 60 j= l,jmax te U

jj = jj - 1

+++++++++++++++ MODI0 ++++++++++++++++++

y(j,k,l) = yy(kmax__w) + (yy(kmax_w) - yy(kk))

y(j,k,l) = w(kk) + yy(kmax_w)

++++++++++++++++++++++++++++++++++++++++

IF(tr_test2 .eq. I) THEN

MOD9

x(j,k,l)=x_U(i) + y(j,k,I)*TAN(GAMA)

x(j,k,l)=x(j,kmax_w,l) + (y(j,k,l) -y(j,kmax_w,I))*TAN(GAMA)

ELSE

TE_sweep = ATAN( (span*TAN(sweep) - 1 + lambda)/span )

Chord_t : (i + y(j,k_max_w+l,l)*(TAN(TE_sweep) - TAN(sweep)) )

******************************************************************

THIS IS A MOD TO EXTEND THE LE_SWEEP INTO

THIS ZERO THICKNESS SECITON

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MOD9

x(j,k,l)=Chord_t + x_U(i) + y(j,k,l)*TAN(sweep) - Chord_r

x(j,k,l)=t(jj) + Chord_t*x_U(npts_U) + y(j,k,l)*TAN(sweep)

ENDIF

z(j,k,l) = 0.0

Continue

C

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This will add a zero thickness section off the Wing Tip chord

******** For the Upper surface *******

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MOD9

i= npts_U - jmax te U + 1

i = npts_U + 1

do 70 j= jmax te U + l,jmax U

do 70 j= jmax_te_U + l,jmax_U + jmax te U

i = i - 1

+++++++++++++++ MODI0 ++++++++++++++++++

y(j,k,l) = yy(kmax_w) + (yy(kmax_w) - yy(kk))


+++++++++++++++++++++++++++++++++++++++

IF(tr_test2 .eq. i) THEN

Chord t = 1.0

x(j,k,l) = Chord_t * x_U(i) + (y(j,k,I)*TAN(GAMA))

ELSE


Chord_t = (i + y(j,kmax_w+l,l)*(T_(TE_sweep) - TAN(sweep)) )

******************************************************************



*W**W**W*W******W**WWW*W*W*WW***W_*WW*WWWWWWWW*WWWWWW**WW**W*WWWW*

x(j,k,l)= Chord_t * x_U(i) + y(j,k,l)*TAN(sweep)

ENDIF

$$$$$$$$$$$$$$$$SSSSSSSSSSSSSSSS_:SSSSSS$$SSSSSSSSSSSSSSS$$SSS$SS$

z(j,k,l) = 0.0

$$$$$$$$$$$SSSSSSSSSSS$$SSSSSSS$_$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

continue

This will add a zero thickness sect:.on off the Wing tip chord

******** For the Lower surface _'******

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counter = 1

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MOD9

do 80 j= jmax_U + l,jmax - jmax te U

do 80 j= jmax U + jmax te U + l,jmax + jmax te U

counter = counter + 1

+++++++++++++++ MODI0 ++++++++++++++++++



++++++++++++++++++++++++++++++++++++++++


Chord_t = 1.0

x(j,k,l) = Chord_t * x_L(counter) + (y(j,k,I)*TAN(GAMA))

ELSE


Chord_t = (i + y(j,kmax_w+l,l)*(TAN(TE_sweep) - TAN(sweep)) )

******************************************************************



**WWWWWW*WWW***W*WWWW*WW**WWWWW**W*WWWWW***WWWWWW**W***WW*WW*WW***

x(j,k,l)= Chord_t * x_L(counter) + y(j,k,l)*TAN(sweep)

ENDIF

$$$$SSSSSSSSSSSS$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

z(j,k,l) = 0.0

continue

This will add a zero thick section off the Wing Tip- Trailing Edge

**** for the lower surface ****

C

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MOD9

i = npts_L - jmax te U

jj = 1

do 90 j= jmax - jmax te U + i, jmax

do 90 j= jmax + jmax te U + I, jmax + jmax_te

i = i + 1

jj = jj + 1

+++++++++++++++ MODI0 ++++++++++++++++++



++++++++++++++++++++++++++++++++++++++++


x(j,k,l)=x_L(i) + y(j,k,I)*TAN(GAMA)

x(j,k,l)=x(j,kmax w,l) + (y(j,k,l) -y(j,kmax_w,I))*TAN(GAMA)

ELSE


Chord_t = (I + y(j,kmax_w+l,l)*(TAN(TE_sweep) - TAN(sweep)) )

******************************************************************


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x(j,k,1)=Chord_t + x_L(i) + y(j,k,1)*T__,l(sweep) - Chord_r

x(j,k,1)=t(jj) + Chord_t*x_U(npts_U) + y(j,k,1)*TAN(sweep)

ENDIF

z(j,k,l) = 0.0

Continue

continue

write grid

WRITE(*,'(a,$)')' ENTER grid FILE NAME: '

READ(*,1000)outfile

WRITE(*,'(a,$)')' IF YOU WANT A b_LTI GRID OUTPUT TYPE i:

READ(*,*)MG

change 'binary' to 'unformatted' to run on CRAY 2 or VAX

OPEN(UNIT=7,FILE=outfile,STATUS='new',

+ form='unformatted')

MOD9

PRINT*,'HEY 2 jmax_te= ',jmax_te

jmax = jmax + jmax_te

PRINT*,'HEY 2 jmax= ',jmax

IDM(1) = jmax

JDM(1) = kmax

KDM(1) = llmax

IF(MG .ne. i) THEN

WRITE (7) jmax, kmax, llmax

WRITE(7) (((X(J,K,L), J=jmax, l,--l), K=l,kmax), L=l,llmax),

+ (((Y(J,K,L), J=jmax, l,-l), K=l,kmax), L=l,llmax),

+ (((Z(J,K,L), J=jmax, l,-l), K=l,kmax), L=l,llmax)

ELSE

NGRID = 1

WRITE(7) NGRID

WRITE(7) (IDM(IGRID),JDM(IGRID),}_M(IGRID),IGRID=I,NGRID)

DO 110 IGRID= I,NGRID

WRITE(7)

(((X(I,J,K),

I=IDM(IGRID),I,-I),J=I,JDM(IGP.ID)),K=I,KDM(IGRID)),

(((Y(I,J,K),

I=IDM(IGRID),I,-I),J=I,JDM(IGkID)),K=I,KDM(IGRID)),

(((Z(I,J,K),

I=IDM(IGRID),I,-I),J=I,JDM(IGLID)),K=I,KDM(IGRID))

CONTINUE

ENDIF

stop

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end

subroutine vinokur(s,lmax,smax,dsle,ds2e)

stretches points on a surface so that a specified spacing

at the boundaries is satisfied. Taken from NASA CR 3313 by

Vinokur (1980).

In this version, 4 distinct iterations are made to better

match the resulting delta-s values to the requested values.

The four iterations are summarized below:

i. delta-s is set equal to the desired value.

2. delta-s from the last iteration is corrected from a Taylor

series expansion.

3. delta-s is calculated from a linear fit between the first two

guesses.

4. delta-s is calculated from a quadratic fit between the first

three guesses, if indeed a quadratic will pass through the

desired value. If it doesn't, it takes the value calculated

after three swipes.

Additionally, this version uses the approximate inverse solution

for y=sin(x)/x and y=sinh(x)/x rather than a Newton iteration. The

approximate solution was also taken from NASA CR 3313.

common

/io/

input,kopy, default

dimension s(200), dl(4,2),d2(4,2)

c

C .. ... .

c for an IRIS 2500,

emax=87.0

c-

c 21

c

c

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21

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22

c22

c

c

dsavg=smax/float(imax-l)

write(*,103)dsavg

PRINT*, 'dsle= ',dsle

PRINT*, 'ds2e= ',ds2e

PRINT*, 'smax= ',smax

dsavg=0.001

dsle=dsavg

call realval(l,l,dsle,q,q,*21,*101)

if(dsle.ge, smax.or.dsle.lt. 0.0)go to 21

dsavg=0.01

write(*,104)dsavg

ds2e=dsavg

call

realval(l,l,ds2e,q,q,*22,*101)

if(ds2e.ge.(smax-dsle).or.ds2e.lt. 0.0)go to 21

if(dsle.eq.0.0.and.ds2e.eq.0.0)then

kase=0

dsle=dsavg

ds2e=dsavg

nlast=4

else if(dsle.eq.0.0)then

kase=l

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nlast=l

write(6,106)

continue

call realval(0,l,slop,no,no,*23,*101)

if(slop.lt.0.0.or.slop.gt.l.0)go to 23

dsle=-slop

else if(ds2e.eq.0.0)then

kase=2

nlast=l

write(6,106)

continue

call realval(0,l,slop,no,no,*24,*101)

if(slop.lt.0.0.or.slop.gt.l.0)go to 24

ds2e=-slop

else

kase=0

nlast=4

end if

dssl=0.0

dss2=0.0

do 6 n=l,nlast

if(n.le.2)then

dsl=dsle-0.5*dssl

ds2=ds2e+0.5*dss2

dl(n,l)=dsl

d2(n,l)=ds2

PRINT*,'dI(I,I)=

PRINT*,'dI(I,2)=

PRINT*,'dI(2,1)=

PRINT*,'dI(2,2)=

else if(n.eq.3)then

',dl(l,l)

',dl(l,2)

',dl(2,1)

',di(2,2)

dsl=-dl(l,2)*(dl(2,1)-dl(l,l))/(dl(2,2)-dl(l,2))+dl(l,l)

PRINT*,'d2(I,I)= ',d2(l,l)

PRINT*,'d2(I,2)= ',d2(i,2)

PRINT*,'d2(2, )= ',d2(2,1)

PRINT*,'d2(2,2)= ',d2(2,2)

ds2=-d2(l,2)*(d2(2,1)-d2(l,l))/(d2(2,2)-d2(l,2))+d2(l,l)

PRINT*,'dsI= ',dsl

PRINT*,'ds2= ',ds2

PRINT*,'nlast= ',nlast

PRINT*,'HELP '

if(dsl.lt.0.0)dsl=0.5*aminl(dl(l,l),dl(2,1))

if(ds2.1t.0.0)ds2=0.5*aminl(d2(l,l),d2(2,1))

dl(n,l)=dsl

d2(n,l)=ds2

else if(n.eq.4)then

denom=-(dl(l,l)-dl(2,1))*(dl(2,1)-dl(3,1))*(dl(3,1)-dl(l,l))

all=dl(2,1)-dl(3,1)

a21=dl(3,1)**2-dl(2,1)**2

a31=dl(2,1)*dl(3,1)*(dl(2,1)-dl(3,1))

al2=dl(3,1)-dl(l,l)

a22=dl(l,l)**2-dl(3,1)**2

a32=dl(3,1)*dl(l,l)*(dl(3,1)-dl(l,l))

al3=dl(l,l)-dl(2,1)

a23=dl(2,1)**2-dl(l,l)**2

a33=dl(l,l)*dl(2,1)*(dl(l,l)-dl(2,1))

bl=(all*dl(l,2)+al2*dl(2,2)+al3*dl(3,2))/denom

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b2=(a21*dl(l,2)+a22*dl(2,2)+a23*dl(3,2))/denom

b3=(a31*dl(l,2)+a32*dl(2,2)+a33*dl(3,2))/denom

disc=(b2*b2-4.*bl*b3)

if(disc.lt.0.0)go to 8

ddl=(-b2+sqrt(disc))/(2.*bl)

dd2=(-b2-sqrt(disc))/(2.*bl)

dd3=dl(3,1)

if(abs(ddi-dd3).it.abs(dd2-dd3))then

dsl=ddl

else

dsl=dd2

end if

denom=-(d2(l,l)-d2(2,1))*(d2(2,1)-d2 3, ))*(d2(3,1)-d2(l,l))

all=d2 (2, i) -d2 (3, i)

a21=d2 (3, i) *'2-d2 (2, i) **2

a31=d2 (2, i) *d2 (3, i) * (d2 (2, i) -d2 (3, I)

a12=d2 (3, i) -d2 (i, i)

a22=d2(l,l)**2-d2(3,1)**2

a32=d2(3,1)*d2(l,l)*(d2(3,1)-d2(l,l)

a13=d2(l,l)-d2(2,1)

a23=d2(2,1)**2-d2(l,l)**2

a33=d2(l,l)*d2(2,1)*(d2(l,l)-d2(2,1)

bl=(ail*d2(l,2)+a12*d2(2,2)+a13*d2(3 2))/denom

b2=(a21*d2(l,2)+a22*d2(2,2)+a23*d2(3 2))/denom

b3=(a31*d2(l,2)+a32*d2(2,2)+a33*d2(3,2))/denom

disc=(b2*b2-4.*bl*b3)

if(disc.le.0.0)go to 8

ddl=(-b2+sqrt(disc))/(2.*bl)

dd2=(-b2-sqrt(disc))/(2.*bl)

dd3=d2(3,1)

if(abs(ddl-dd3).it.abs(dd2-dd3))then

ds2=ddl

else

ds2=dd2

end if

if(dsl.lt.0.0.or.ds2.1t.0.0)go to 8

end if

calculate constants

s0=smax/float(imax-l)/dsl

sl=smax/float(Imax-l)/ds2

b=sqrt(s0*sl)

a=sqrt(s0/sl)

if(kase.eq.l)then

b=sl

else if(kase.eq.2)then

b=s0

end if

calculate x based on value of B

if(b-l.)i,2,3

x is real

if(b.lt.0.26938972)then

pi=4.*atan(l.)

x=pi*(l. -b + b**2

* + 6.794732 b*'4 -13.205501 b*'5

else

- (l.+pi**2/6.)*b**3

+ ii.726095 b*'6)

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c

c

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c=l. -b

x= sqrt

(6.*c)*

(i.

* +0.15 c

*

-0. 053337753 c*'4

end if

go to 4

+ 0.057321429 c*'2

+ 0.07584513_*c*'5)

+0.048774238 c*'3

x is zero

x=O.

go to 4

x is imaginary

if(b.lt.2.7829681)then

c=b-l.

x= sqrt(6.*c)*(l.

*

-0.15 c + 0.057321429 c*'2

* +0.0077424461 c*'4 -0.0010794123 c*'5)

else

v=alog(b)

w=l./b - 0.028527431

x= v + (l.+l./v)*alog(2.*v) -0.02041793

* + 0.24902722 w + 1.9496443 w*'2

* - 2.6294547 w*'3 + 8.56795911 w*'4

end if

-0.024907295 c*'3

distribute points along boundary

continue

if(kase.eq.l.or.kase.eq.2)then

s(l ) = 0.0

s(imax) = smax

do 9 i=2,1max-i

j= imax+l-i

xi=float(i-l)/(imax-l)

if(b.gt.l. OOOl)then

ul=l. + tanh(x/2.*(xi-l.))/tanh(x/2.)

else if(b.lt.O.9999)then

ul=l. + tan (x/2.*(xi-l.))/tan (x/2.)

else

ul= xi*(l.-.5*(b-l.)*(l.-xi) _(2.-xi))

end if

u2=sinh(xi*x)/sinh(x)

if(kase.eq.l)then

fact=abs(dsle)

s(j) = ( (l.-fact)*(l.-ul)

else if(kase.eq.2)then

fact=abs(ds2e)

s(i) = ( (l.-fact)* ul

end if

continue

else

do 5 i=l,lmax

xi=float(i- )/float(Imax-l)

cnum=x*(xi-0.5)

cden=x/2.

if(b.lt.O.9999)then

cc=tan(cnum)/tan(cden)

u=0.5*(l.+cc)

+ fact*(l.-u2) ) *smax

+ fact* u2 ) *smax

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else if(b.ge.O.9999.and.b.le.l. OOOl)then

u=xi*(l.+2.*(b-l.)*(xi-O.5)*(l.-xi))

else if(b.gt.l. OOOl)then

cc=tanh(cnum)/tanh(cden)

u=0.5*(l.+cc)

end if

s(i)=u*smax/(a+(l.-a)*u)

end if

if(imax.ge.4)then

dssl=( -s(4) +4.*s(3) -5.*s(2)

dss2=(2.*s(imax)-5.*s(imax-l)+4.*s(imax-2)

end if

+2.*s(1))

-s(imax-3))/2.

esl=s(2)-s(1)

es2=s(imax)-s(imax-l)

if(n.ne.4)then

dl(n,2)=esl-dsle

d2(n,2)=es2-ds2e

end if

continue

esmin= l.Oe+08

esmax=-l.Oe+08

do 7 j=2,1max

stmp=s (j) -s (j-l)

if(stmp.lt.esmin)then

jnj =j

esmin=stmp

end if

if(stmp.gt.esmax)then

jxj =j

esmax=stmp

end if

continue

write(6,105)esl,es2,jnj-i jnj,esmin, jxj-l,jxj,esmax

return

format(/,6x, 'enter delta s at beginning of arclengt

Documents

NASA TM-108852: Parametric study on laminar flow for finite wings at supersonic speeds